
Journal of Convex Analysis 22 (2015), No. 3, 673685 Copyright Heldermann Verlag 2015 Ball Proximinal and Strongly Ball Proximinal Spaces PeiKee Lin Dept. of Mathematics, University of Memphis, Memphis, TN 38152, U.S.A. pklin@memphis.edu Wen Zhang School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China wenzhang@xmu.edu.cn Bentuo Zheng Dept. of Mathematics, University of Memphis, Memphis, TN 38152, U.S.A. bzheng@memphis.edu [Abstractpdf] Let $Y$ be an $E$proximinal (respectively, a strongly proximinal) subspace of $X$. We prove that $Y$ is (strongly) ball proximinal in $X$ if and only if for any $x\in X$ with $(x+Y)\cap B_X\ne\emptyset$, $(x+Y)\cap B_X$ is (strongly) proximinal in $x+Y$. Using this characterization and a smart construction, we obtain three Banach spaces $Z\subset Y\subset X$ such that $Z$ is ball proximinal in $X$ and $Y/Z$ is ball proximinal in $X/Z$, but $Y$ is not ball proximinal in $X$. This solves a problem raised by P. Bandyopadhyay, BorLuh Lin and T.S.S.R.K. Rao [{\em Ball proximinality in Banach spaces,} in: Banach Spaces and Their Applications in Analysis (Oxford/USA, 2006) B. Randrianantoanina et al (eds.) Proceedings in Mathematics, de Gruyter, Berlin (2007) 251264]. Keywords: Ball proximinal, strongly ball proximinal, Eproximinal. MSC: 46B20, 41A50 [ Fulltextpdf (131 KB)] for subscribers only. 